Let $Y$ be a smooth complex projective curve of genus two, $X$ a Galois cover of degree two of $Y$ and $K$ the canonical divisor of $X$. Let $i$ be the involution of $X$ over $Y$. Can one find two points $P$ and $P'$ on $X$ such that

$$ 2P' = 3P + 3i(P) - K \, $$

and $P$ is not conjugate to $i(P)$ (note that $X$ is hyperelliptic)?

(Thanks to abx for his answer to my previous question).